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1

In order to motivate examples in the first class in congruence theory, my teacher remarked that the beginning chapters of the Holy Bible mathematically said entail the following: "Let the days of the week be congruent modulo seven."
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UnknownJun 12 '10 at 17:07

17

Why did a question with so much positive feedback get closed?
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RomeoNov 28 '10 at 23:21

@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing.
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Qiaochu YuanNov 12 '13 at 3:03

97 Answers
97

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

The attitude adopted in this book is
that while we expect to get numbers
out of the machine, we also expect to
take action based on them, and,
therefore we need to understand
thoroughly what numbers may, or may
not, mean. To cite the author's
favorite motto,

“The purpose of computing is insight,
not numbers,” although some people
claim,

“The purpose of computing numbers is
not yet in sight.”

There is an innate risk in computing
because “to compute is to sample, and
one then enters the domain of
statistics with all its
uncertainties.”

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure... The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us." - Andre Weil

For general continuous curves, it's not that a simple proof [of the Jordan curve theorem] is not possible, it's that it's not desirable. The true content of the result is homology theory, which proves the separation result in n dimensions. There are special proofs in 2D that are simpler, but every such proof that I have seen feels like a one-night stand.

Without pretty ßs: Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters.

Taken out of context, this would seem to be accurate (the concept of evil with respect to higher categories), but one must remember that Poincar\'e was not against axiomatic set theory per se, but axiomatic theories in general.
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Harry GindiNov 30 '09 at 6:33

12

Actually this is not a quote. It is attributed to Poincare in various sources, but it is quite likely that this is a misinterpretation of something that he actually wrote (something to the effect that the diseases of set theory (such as Russell's paradox, for example) will one day be overcome). See J. Gray, Did Poincaré say ``Set theory is a disease''?, Math. Intell. 13 (1991), 19--22
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Franz LemmermeyerFeb 15 '10 at 16:21

This one has to do with the quote by Rota that appears in the first post of C. Siegel:

" The essence of Mathematics is proving theorems and so, that is what mathematicians do: they prove theorems. But to tell the truth, what they really want to prove once in their lifetime, is a lemma, like the one by Fatou in Analysis, the lemma of Gauss in Number Theory, or the Burnside-Frobenius lemma in Combinatorics.

Now what makes a mathematical statement a true lemma? First, it should be applicable to a wide variety of instances, even seemingly unrelated problems. Secondly, the statement should, once you have seen it, be completely obvious. The reaction of the reader might well be one of faint envy: Why haven't I noticed this before? And thirdly, on an esthetic level, the lemma including its proof should be beautiful!"

"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact opposite!" -- Paul Dirac (some people attribute it to Franz Kafka!?)

Ken Ribet once told me the story of how he was sent a freebie book "for possible use in your undergraduate classes" that he looked at and decided he didn't want, so took it to the second hand bookstore in his lunch break, sold it, and bought lunch with the proceeds. On the way back to the math department he realised he'd turned theorems into coffee.
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Kevin BuzzardNov 29 '09 at 22:42

43

A comathematician is a device for turning cotheorems into ffee. A cotheorem is of course what one deduces from a rollary.
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Saul GlasmanNov 30 '09 at 19:02

A good definition can occassionally be worth a thousand proofs, I think, but frankly, elementary differential geometry seems to me to fit the “desert of definitions” pretty well. Not that there is anything wrong in that. Some times you have to suffer loads of boring definitions in order to see the depth and beauty of the subject. (And besides, deserts can be beautiful too.)
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Harald Hanche-OlsenDec 8 '09 at 18:19

Farkas Bolyai to his son Janos, speaking about attempts to study Euclid's Vth postulate on parallel lines:

"You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of the parallels alone... I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction.... I turned back when I saw that no man can reach the bottom of the night. I turned back unconsoled, pitying myself and all mankind.

I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness - aut Caesar aut nihil."

In the biographical piece on Grothendieck a couple of years ago in the Notices <<http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf>> the author says "One thing Grothendieck said was that one should never try to prove anything that is not almost obvious". It's not a quote, but it is a nice succinct way of putting his 'nut' analogy given above.

(My poor translation: For how long will young people be forced to listening or memorizing during whole days? When will they be allowed time to ponder on this mass of knowledge, to coordinate the multitude of unconnected propositions, of unrelated calculations? … Instead, they are carefully taught truncated theories, loaded with unnecessary reflections, while omitting the most brilliant propositions of algebra…)

“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”- Godfrey Harold Hardy